Bernoulli Number Calculator – B Sequence

Bernoulli numbers arise from the generating function t/(eᵗ−1): expanding the series gives B₀ = 1, B₁ = −1/2, B₂ = 1/6, B₃ = 0, B₄ = −1/30, B₆ = 1/42, alternating in sign for even indices while every odd index past one vanishes. The calculator returns exact fractions (B₁₀ = 5/66, B₁₂ = −691/2730) with decimal equivalents, using the recurrence Σ C(n+1,k)Bₖ = 0 solved in rational arithmetic — no floating drift. Their fame comes from two outposts: power sums follow Faulhaber’s formula Σₖ₌₁ⁿ k^p = (Bₚ₊₁ polynomial in n)/(p+1), e.g. Σk = n(n+1)/2 built from B₁; and the tangent expansion tan x = x + x³/3 + 2x⁵/15 + 17x⁷/315 has numerators drawn from even Bernoulli numbers. They also seed the Riemann zeta values ζ(2n).

1. 1

   Enter the index n

   Any non-negative integer

2. 2

   Solve the recurrence

   Rational arithmetic produces the exact fraction

3. 3

   Read the value

   B₄ = −1/30 ≈ −0.0333, zeros flagged for odd n > 1

4. 4

   Browse the sequence

   The table around your index is displayed

• 1

  Odd Bernoulli numbers beyond B₁ are all zero

• 2

  Signs alternate on even indices: B₄ = −1/30, B₆ = +1/42

• 3

  Exact fractions matter — B₁₂ = −691/2730 defies decimal guesswork

• Using B₁ = +1/2 (the "second convention") in formulas expecting −1/2

• Expecting nonzero odd terms after B₁

• Decimal rounding destroying the delicate cancellations in recurrences

• Forgetting B₀ = 1 anchors the entire sequence